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Properties

Label 2-960-40.29-c1-0-6
Degree $2$
Conductor $960$
Sign $0.450 - 0.892i$
Analytic cond. $7.66563$
Root an. cond. $2.76868$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 3-s + (2.18 + 0.456i)5-s + 0.913i·7-s + 9-s + 3.58i·11-s + 0.913·13-s + (−2.18 − 0.456i)15-s − 3.58i·17-s + 4i·19-s − 0.913i·21-s + (4.58 + 1.99i)25-s − 27-s + 7.84i·29-s − 5.29·31-s − 3.58i·33-s + ⋯
L(s)  = 1  − 0.577·3-s + (0.978 + 0.204i)5-s + 0.345i·7-s + 0.333·9-s + 1.08i·11-s + 0.253·13-s + (−0.565 − 0.117i)15-s − 0.868i·17-s + 0.917i·19-s − 0.199i·21-s + (0.916 + 0.399i)25-s − 0.192·27-s + 1.45i·29-s − 0.950·31-s − 0.623i·33-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 960 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.450 - 0.892i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 960 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.450 - 0.892i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(960\)    =    \(2^{6} \cdot 3 \cdot 5\)
Sign: $0.450 - 0.892i$
Analytic conductor: \(7.66563\)
Root analytic conductor: \(2.76868\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{960} (289, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 960,\ (\ :1/2),\ 0.450 - 0.892i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.26193 + 0.776513i\)
\(L(\frac12)\) \(\approx\) \(1.26193 + 0.776513i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 + T \)
5 \( 1 + (-2.18 - 0.456i)T \)
good7 \( 1 - 0.913iT - 7T^{2} \)
11 \( 1 - 3.58iT - 11T^{2} \)
13 \( 1 - 0.913T + 13T^{2} \)
17 \( 1 + 3.58iT - 17T^{2} \)
19 \( 1 - 4iT - 19T^{2} \)
23 \( 1 - 23T^{2} \)
29 \( 1 - 7.84iT - 29T^{2} \)
31 \( 1 + 5.29T + 31T^{2} \)
37 \( 1 + 7.84T + 37T^{2} \)
41 \( 1 - 6T + 41T^{2} \)
43 \( 1 - 7.16T + 43T^{2} \)
47 \( 1 + 6.92iT - 47T^{2} \)
53 \( 1 - 2.55T + 53T^{2} \)
59 \( 1 - 7.58iT - 59T^{2} \)
61 \( 1 - 10.5iT - 61T^{2} \)
67 \( 1 - 15.1T + 67T^{2} \)
71 \( 1 + 6.92T + 71T^{2} \)
73 \( 1 - 12iT - 73T^{2} \)
79 \( 1 - 5.29T + 79T^{2} \)
83 \( 1 + 11.1T + 83T^{2} \)
89 \( 1 + 2T + 89T^{2} \)
97 \( 1 - 7.16iT - 97T^{2} \)
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   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−10.24204719903077463943567830599, −9.424237526797881476820647567803, −8.744704421404828673569083867807, −7.32647339353160700048546226812, −6.84091257430926103449703202265, −5.65954579883810340481338747888, −5.25769583559390786914450888689, −3.99799179509554933589654530214, −2.57131597737954943385888648256, −1.48762224891828980202800188457, 0.792221961027751375156753631902, 2.17751550835024770514961136568, 3.58626220654420714593862702121, 4.70836997104375175975423314991, 5.76663801557420136190861758203, 6.18191725566532685177017514086, 7.21393737166478175867526723201, 8.323273839823984572488104234752, 9.126662099871523633126277164924, 9.916642553786649324049285847755

Graph of the $Z$-function along the critical line